How can I use the success and failure data to estimate parameters of a Dirichlet distribution?
Dirichlet is the Multi Variate version of the Beta distribution. In general, these distributions can be thought to model the probability of modelling a probability distribution. 

The support Dirichlet distribution is defined as follows:

$$ S_K = \{ x:0 \leq x_k \leq 1, \sum_{k=1}^K x_k=1 \} $$

and the PDF is defined as:

$$Dir(x|\alpha) = \frac{1}{B(\alpha)} \prod_{k=1}^Kx_k^{\alpha_k-1}$$

where $B(\alpha)$ is the beta function of $K$ variables:

$$B(\alpha) = \frac{\prod_{k=1}^K \tau(\alpha_k)}{\tau(\sum_{k=1}^K \alpha_k)}$$

and the resultant point estimates are (Define $\sum_{k=1}^K \alpha_k = \alpha_0)$:

$$\mu(x_k) = \frac{\alpha_k}{\alpha_0}$$
$$\sigma^2(x_k) = \frac{\alpha_k(\alpha_0-\alpha_k)}{\alpha_0^2(\alpha_0+1)}$$
$$mode[x_k] = \frac{\alpha_k-1}{\alpha_0-1}$$

Beta distribution is the special case where $k=2$

Clearly, when you run an experiment a large number of times, the success of each $k$ will approach towards its expected value i.e if you define your random variables as $x_k = \frac{N_{k}}{N}$ where $N$ is the total number of trials and $N_k$ is the success of the $k$ th term, it clearly satisfies the support of the Dirichlet Distribution and hence you can use 

$$\frac{\alpha_{k}}{\alpha_0} = \frac{N_{k}}{N}$$

This is assuming that the experiment follows Dirichlet Distribution.

(Taken in parts from A Probabilistic Approach to ML)